Scheme-Independent Series Expansions at an Infrared Zero of the Beta Function in Asymptotically Free Gauge Theories

Abstract

We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, having an infrared (IR) zero in the beta function at ${\alpha }_{IR}$. We present general formulas for scheme-independent series expansions of quantities, evaluated at ${\alpha }_{IR}$, as powers of an $N_f$-dependent expansion parameter, ${\Delta }_f$. First, we apply these to calculate the derivative $d\beta /d\alpha$ evaluated at ${\alpha }_{IR}$, denoted ${\beta }_{IR}^{\prime }$, which is equivalent to the anomalous dimension of the $\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })$ operator, to order ${\Delta }_f^4$ for general $G$ and $R$, and to order ${\Delta }_f^5$ for $G=\mathrm{SU}(3)$ and fermions in the fundamental representation. Second, we calculate the scheme-independent expansions of the anomalous dimension of the flavor-nonsinglet and flavor-singlet bilinear fermion antisymmetric Dirac tensor operators up to order ${\Delta }_f^3$. The results are compared with rigorous upper bounds on anomalous dimensions of operators in conformally invariant theories. Our other results include an analysis of the limit $N_c\to \infty$, $N_f\to \infty$ with $N_f/N_c$ fixed, calculation and analysis of Padé approximants, and comparison with conventional higher-loop calculations of ${\beta }_{IR}^{\prime }$ and anomalous dimensions as power series in $\alpha$.

26 pages latex, 3 figures

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